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On scaling of mass entrainment in separated shearlayers: the footprint of the incoming boundary layer

Francesco Stella, Nicolas Mazellier† and Azeddine Kourta

Univ. Orléans, INSA-CVL, PRISME, EA 4229, F45072, Orléans, France

(Received xx; revised xx; accepted xx)

We experimentally investigate the effects on scaling of separating/reattaching flows of the ratioδe/h, where δe is the thickness at separation of the incoming boundary layer and h the char-acteristic cross-stream scale of the flow. In the present study, we propose an original approachbased on mean mass entrainment, which is the driving mechanism accounting for the growth ofthe separated shear layer. The focus is on mass transfer at the Turbulent/Non-Turbulent Interface(TNTI). In particular, the scaling of the TNTI, which is well documented in turbulent boundarylayers, is used to trace changes in the scaling properties of the flow. To emphasise the influenceof the incoming boundary layer, two geometrically similar, descending ramps with sizeablydifferent heights h but fundamentally similar values of δe are compared. The distribution in spaceof the TNTI highlights a sizeable footprint of the incoming boundary layer on the separated flow,the scaling of which results of the competition between h and δe. On the basis of a simplemass budget within the neighbourhood of separation, we propose to model this competitionby introducing the scaling factor Ch,δ = 1 + δe/h. With this model, we demonstrate that therelationship between shear layer growth and mass entrainment rates established for free shearlayers (i.e. δe = 0) might be extended to flows where δe/h > 0. Since many control systemsrely on mass entrainment to modify separation properties, our findings suggest that the parameterδe/h needs to be taken into account when choosing the most relevant strategies for controllingor predicting separating/reattaching flows.

1. Introduction

Geometry-triggered separating/reattaching flows are common in industrial applications. Theirundesirable effects, including degraded aerodynamic performances, increased vibrations andnoise, motivate the great efforts which have been dedicated to investigating these flows (seefor example the review by Nadge & Govardhan (2014)). On the backward-facing step (BFS)and other prototypical salient-edge bluff bodies, separation is fixed by geometry. Downstreamof separation, the flow develops into a wide shear layer (Simpson (1989)), which grows until itimpinges the wall at the reattachment point, several step heights downstream of the BFS. Theregion of the flow lying between the wall and the shear layer is called recirculation region, dueto the reversed direction of the main velocity component. The local depression induced by therecirculation region is at the core of many negative effects of flow separation, and in particular ofthe sizeable drag increase which it usually produces. Accordingly, Roshko & Lau (1965) showthat the length of the recirculation region LR, measured as the streamwise distance betweenthe separation point and the reattachment point, is the characteristic length scale of the reduced

streamwise pressure distributions past a wide set of different bluff bodies, at least if separationis geometrically fixed. Since, by definition, LR is the scale of shear layer development, it isthen generally admitted that interacting with the shear layer to artificially tune LR might be aneffective strategy to change the pressure distribution in separating/reattaching flows, and hence

† Email address for correspondence: [email protected]

http://arxiv.org/abs/1811.05025v1

2 F. Stella, N. Mazellier and A. Kourta

to control drag (Chun & Sung (1996), Berk et al. (2017), Stella et al. (in press)). Unfortunately,drag control solutions based on this approach have proven hard to scale up from laboratorymodels to full-size industrial applications. This problem appears to be largely linked to thegreat sensitivity of the spreading the separated shear layer to a number of flow and geometryparameters, such as free-stream turbulence (Adams & Johnston (1988b)), the shape of the bluffbody (Ruck & Makiola (1993)) or, to some extent, its expansion ratio ER (Nadge & Govardhan(2014)). In this instance, the boundary layer upstream of separation, when present, deservesparticular attention, because it provides the initial conditions of shear layer development (forexample, see the discussion of momentum thickness θ in Chun & Sung (1996)). As such, it canaffect the separating/reattaching flow in many different ways, one classical example being itslaminar/turbulent state (Armaly et al. (1983)). Interestingly, the incoming boundary layer ap-pears to have macroscopic effects on the streamwise pressure distribution induced by a separatedflow. Tani et al. (1961), Westphal et al. (1984) and in particular Adams & Johnston (1988a) showthat pressure recovery at reattachment depends on the ratio δe/h, where δe is the full thicknessof the boundary layer at separation and h is the characteristic cross-stream scale of the bluffbody (typically, the height of the BFS). More in details, when δe/h > 0.3 the streamwisewall-pressure distribution progressively deviates from its pseudo-universal form observed byRoshko & Lau (1965): the maximum reduced wall-pressure coefficient Cp,max decreases forincreasing values of δe/h, eventually reaching a minimum value imposed by ER. These resultsappear to be consistent with some of the key concepts of the theory of Nash (1963), whichpredicts that the wall-pressure coefficient at reattachmentCp,r should decrease as the thickness ofthe shear layer at separation increases. In spite of the difference between Cp,r and Cp,max, suchagreement suggests that the incoming boundary layer influences the initial development of theseparated flow. Depending on the value of δe/h, this footprint might be more or less persistent,and possibly propagate up to reattachment. In other words, separating/reattaching flows appearto generally depend on both h and δe, with the relative strength of the two characteristic lengthscales changing across the velocity field (see for example Song & Eaton (2003, 2004)). Thisstrongly suggest that, as other multi-scale flows such as plane wakes(Wygnanski et al. (1986)),separating/reattaching flows cannot be considered fully self-similar in a general way, unlike freeshear layers (Pope (2000)).

Lack of self-similarity might have far reaching consequences, because it questions the com-mon assumption that assimilates the separated shear layer to a free shear layer (see Dandois et al.

(2007) and references therein). This has proven a useful hypothesis in the inverstigation ofseparating/reattaching flows. Among other advantages, indeed, it allows us to approximate thecross-stream velocity profile with an error function (Chapman et al. (1958); Tanner (1973))and hence to provide a scaling for the main mean shear component ∂U/∂y. This is a veryimportant result, because mean shear has a key role in amplificating shear layer instabilities,in enhancing turbulent production and in many more (often detrimental) phenomena that areusually companions of separation. In this respect, lack of self-similarity makes the analysis ofthese behaviours much harder, because it implies that, for δe/h > 0.3, the scaling of ∂U/∂y alsodepends on δe and changes in the streamwise direction in a non-trivial way. For these reasons,investigating the nature of the influence of δe/h at separation seems of great theoretical andpractical interest, with possible implications for modeling of separating/reattaching flows andtheir control in full-scale applications. In this work, we contribute to this effort by studying theeffect of the parameter δe/h on a prototypical separating/reattaching flow.

The first issue to be addressed relates to our capability in identifying and assessing the footprintof the BL. One classical approach might rely on the very lack of self-similarity of the flow.In this view, the profile of, say, mean streamwise velocity is expected to progressively mutatefrom the log-law typical of boundary layers, to the error function profile that is characteristic offree shear layers. Then, identifying the footprint of the boundary layer comes down to mapping

Boundary layer footprint on separated flows 3

the regions of the flow in which δe dominates the scaling laws. This kind of analysis can beattempted with some success (Song & Eaton (2004)), but local scaling changes are more visibleif the characteristic scales of the flow are sizeably different. Unfortunately, results reported byAdams & Johnston (1988a) suggest that our analysis is most relevant when δe and h are similar(e.g. δe/h ∈ (0.3, 1)): then, directly investigating local scaling of velocity profiles is not anefficient tool to track the footprint of the boundary layer. In this work we propose an originalapproach to solve this problem, based on the analysis of mass entrainment. Mass entrainment hasthe major advantage of being an integral quantity: it does not rely on self-similarity assumption,or any local δe/h effects, and gives a global picture of the footprint of the boundary layer on theseparated flow. In addition, it is well known that mass entrainment drives the growth of turbulentboundary layers (Chauhan et al. (2014b,c)) as well as the development of free shear layers (Pope(2000)). In their recent work, Stella et al. (2017) quantitatively show that this is also the casein a separated shear layer. Anyway, boundary layers and shear layers grow (i.e. entrain externalfluid) in sizeably different ways: then, mass entrainment is also likely to be a powerful tracerof differences between the two categories of flows. A further consideration in favour of ourapproach stems from the comparison of free shear layers and separated shear layers. Generallyspeaking, these flows differ for their geometrical boundary conditions and, depending on δe/h,for their initial conditions. However, the role of mass entrainment in their development is similar(Stella et al. (2017)). This suggests that mass entrainment might be a robust descriptor of thephysical behaviour of a separated flow, regardless to the value of δe/h and hence to the accuracyof the free shear layer approximation. Significantly, the recent papers by Berk et al. (2017) andStella et al. (in press) indicate that this might even be the case if the separated shear layer isforced with an external control action. In the light of these findings, mass entrainment stands outas a very promising tool to identify the footprint of the incoming boundary layer.

Of course, mass entrainment in turbulent flows is not a new topic. In this respect, manyworks have highlighted the importance of the Turbulent/Non-Turbulent Interface (TNTI)in transfer of mass, momentum and energy from the free-stream to the turbulent region ofthe flow. Research has focused on canonical turbulent flows such as jets (Westerweel et al.

(2009), da Silva & dos Reis (2011)), wakes (Bisset et al. (2002)) and in particular turbulentboundary layers (Corrsin & Kistler (1955), Fiedler & Head (1966), Hedley & Keffer (1974),Chauhan et al. (2014b), Chauhan et al. (2014b), Borrell & Jiménez (2016) among others). Afirst effort to investigate the TNTI in non-canonical flows is reported by Stella et al. (2017),suggesting that some of the lessons learned on canonical flows can be directly extended to theTNTI of separated flows. Some aspects of the instantaneous, local behaviour of the TNTI are stillunder debate, in the first place concerning the nature of its dominant transfer mechanism (seefor example Corrsin & Kistler (1955), Townsend (1966), Taveira et al. (2013) and Mistry et al.

(2016)). Anyway, it is generally agreed that the statistical behaviour of the TNTI respects flowself-similarity. In particular, it is known since the seminal work of Corrsin & Kistler (1955) thatin a turbulent boundary layer the instantaneous TNTI location above the wall approximatelyfollows a gaussian distribution, scaled by the thickness of the boundary layer δ (Chauhan et al.

(2014c) and references therein). These findings can be very useful for our present investigation.In fact, although the incoming boundary layer might be subjected to a weak pressure gradient(see Kourta et al. (2015) among others), it does not seem unreasonable to assume that its TNTIwill still be approximately gaussian distributed, and scaled by δ. It can also be expected that, asthe separated flow departs from self-similarity, this characteristic gaussian TNTI signature willprogressively fade into a different distribution. Then, the first objective of the present study is toinvestigate the local distribution of the TNTI over a separated flow. This should allow to extendthe work of Stella et al. (2017) on the TNTI of a separating/reattaching flow, while contributingat identifying the footprint of the incoming boundary layer.

Our second objective consists in investigating how such footprint modifies the development of


the flow in the region downstream of separation. In particular, we are interested in understandinghow the behaviour of the shear layer is affected by the variation of δe/h. This certainly is a vastsubject, that cannot be exhausted in a single study. Anyway, a question of primary importance thatcan be addressed with reasonable effort concerns the scaling of the main mean shear component∂U/∂y. Indeed, as already stated, the introduction of δe as a second characteristic scale ofthe separated flow might have far reaching consequences on how the shear layer shapes manyproperties of the entire flow.

As a third contribution, we use mass entrainment to investigate whether δe/h affects the freeshear layer analogy. Under this hypothesis, indeed, the mean spreading rate of the separated shearlayer is considered proportional to the sum of mean mass entrainment rates at its boundaries Pope(2000). This has been verified by Stella et al. (2017) even with a relatively high value of δe/h,but the possible effects of the incoming boundary layer on entrainment rates has not yet beenanalysed. Anyway, if δe/h modifies the velocity field (e.g. ∂U/∂y), it can be expected that itmight also have a sizeable impact on the scaling of mass entrainment rates, and possibly on thefree shear layer analogy.

The paper is structured as follows: § 2 presents the experimental set-up; § 3 deals with TNTIstatistics and with the identification of the footprint of the boundary layer; the development ofthe separated shear layer and its scaling are treated at § 4; § 5 analyses the relationship betweenshear layer growth and mass entrainment; conclusions are given at § 6. In the remainder of thepaper, we will adopt the Reynolds decomposition of the velocity field and its standard notation.For example, the instantaneous streamwise velocity component u will be expressed as:

u = U + u′, (1.1)

where U and u′ are the mean and fluctuating streamwise velocities, respectively. The sameconvention applies to the wall-normal velocity component v. The symbol ∗ is used to indicatenormalisation of lengths on LR, or of mass fluxes on ρU∞h, where ρ is density of air and U∞ isfree-stream velocity.

2. Experimental set-up

The reference geometry for this research is a descending, 25◦ ramp that causes the massiveseparation of an incoming turbulent boundary layer (see figure 1). This section presents the twoexperimental models as well as the measurements techniques used in this study.

2.1. Experimental models

Experiments were carried out on two geometrically similar ramps, spanning two differentstep heights but with essentially similar values of δe. This allows to study the effect on theflow of sizeably different values of the ratio δe/h. The first experimental model is the so-calledGDR ramp, which was used in previous studies such as Debien et al. (2014) and Kourta et al.

(2015). The reader is referred to these works for a complete description of the model, the mainproperties of which are summarised in table 1. The GDR step height is h = 100mm. The secondexperimental model was already presented in details in Stella et al. (2017). For simplicity, in theremainder of this paper it will be indicated as the R2 ramp. The R2 ramp has a step height h =30mm, but values of the expansion ratio ER and of the aspect ratio AR are comparable to thoseof the GDR ramp (see table 1). On the contrary, the ratio δe/h is about three times higher than onthe GDR ramp. Together, the two experimental models allow us to cover almost 1.5 decades ofthe similarity parameter Reh = U∞h/ν, where U∞ is a reference velocity and ν is the kinematicviscosity of air.


h/mm AR ER Reh/104 δe/h

√u′2/U∞ [%]

GDR 100 20 1.11 7 to 35 ≈ 0.3 <0.3R2 30 17 1.06 1 to 8 ≈ 0.85 <0.25

TABLE 1. Geometry-based parameters of the GDR ramp, and of the R2 ramp used in Stella et al. (2017).h is the height of the step, and AR and ER are the aspect ratio and the expansion ratio, respectively. Theratio

√u′2/U∞, evaluated within the free stream, is used to measure residual turbulent intensity in the two

facilities.

h

LRFIGURE 1. Outline of the mean separating/reattaching ramp flow investigated in this study. X and Y arerespectively the streamwise axis and the wall-normal axis of a reference system centered on the salient edgeat which the flow separates. Recirculation Region Interface. The three characteristic length scalesof the flow (δe, h and LR) are also reported. U∞ is measured above the upper edge of the ramp.

2.2. Measurement devices

The investigation of the massive turbulent separation is mainly based on Particle Image Ve-locimetry (PIV). Since the mean flow is bidimensional (see Kourta et al. (2015) and Stella et al.

(2017)), 2D-2C PIV is relevant for the purposes of this study. On the GDR ramp, PIV images areobtained with two LaVision VC-Imager cameras, synchronised with a double pulse Nd:Yag laser(wavelength 532nm, rated 2×200 mJ). Each camera is equipped with a Nikon Nikkor 105 lens,yielding an image resolution of about 118 µmpx−1 on a field of view of 4.6h× 2h. The flow isseeded with olive oil droplets of mean diameter dp = 1 µm. Their characteristic time responseis low enough for them to accurately trace all scales of the flow (see Stella et al. (2017)). Foreach Reh, 2000 particle image pairs are recorded at midspan, with an aquisition rate of 2Hz.Then, image pairs are correlated with the multipass, FFT algorithm of the Davis 8.3 softwaresuite. The size of the final correlation window is 32× 32 px2 with 50% overlapping, yieldinga space resolution ∆/h ≈ 0.03, which is adequate for an investigation of the mean field. Aftercorrelation, the vector fields yielded by the two cameras are merged, for a total field of viewof 6h × 2h. On the R2 ramp, PIV images are recorded with one LaVision VC-Imager camera,equipped with Zeiss 50mm ZF Makro Planar T* lens, which provides a camera resolution of78 µmpx−1 and an exploitable field of view of 6h x 2.5h. Laser setting were identical as onthe GDR ramp. For each tested Reh, the R2 ramp database provides 2 fields of view, partiallyoverlapping. For each field of view, a set of 2000 PIV images is available. Instantaneous imagesof differents sets are not correlated, but field statistics can be merged to give a total field of


view of approximately 9h × 2.5h, covering the entire mean recirculation region. PIV imagesare correlated with the multipass, GPU direct correlation algorithm of the LaVision Davis 8.3software suite. The size of the final interrogation window is 16× 16 px2, with 50% overlapping.The spatial resolution is ∆/h ≈ 0.04.

The thickness of the boundary layer at separation δe is measured with a single-component hot-wire probe (Dantec 55P15), driven in constant-temperature mode by a Dantec Streamline 90N10frame. The sensing length of the probe is ℓw = 1.25mm. A discussion of the filtering effectdue to ℓw is provided by Philip et al. (2013). It is stressed that the value of δe considered here isthe full thickness of the boundary layer, i.e. the distance at which U ≈ U∞. The turbulent stateof the incoming boundary layer is often evaluated with the parameter Reθ = θU∞/ν, whereθ is momentum thickness (see for example Song & Eaton (2004)). To allow comparison withStella et al. (2017), Reθ is assessed from hot-wire measurements at a reference section placedat x/h = −9. In addition, auxiliary sets of 2000 PIV images of the incoming boundary layerare recorded at this reference section. For both ramps, characteristics of the PIV set-up andcorrelation settings are the same as detailed above for the main PIV fields. This gives∆/δ ≈ 0.09on the GDR ramp and∆/δ ≈ 0.04 on the R2 ramp, where δ is the thickness of the boundary layerat x/h = −9. Figure 2 shows that velocity profiles of the reference turbulent boundary layer,obtained both from hot-wire and PIV measurements, agree with DNS data by Schlatter & Örlü(2010) sufficiently well to confidently consider larges-scale properties such as θ.

2.3. Estimating the recirculation length

Since the mean topology of the separated flow is substantially comparable across experiments,figure 1 reports a generic sketch of the mean streamwise velocity field. The incoming boundarylayer separates at the upper edge of the ramp, giving origin to the separated shear layer and therecirculation region. The external boundary of the recirculation region is the mean separationline. For consistency with Stella et al. (2017), we will indicate it as the Recirculation RegionInterface (RRI). The RRI is defined by the isoline U = 0 on the mean streamwise velocity field(see Kourta et al. (2015)). In principle, the reattachment point can be identified as the last point ofthe mean RRI (see for example Le et al. (1997) or Kourta et al. (2015)). Unfortunately, in mostof our PIV datasets a thin region in close proximity of the wall was unexploitable, due to laserreflections. Then,LR was estimated as follows: the RRI was approximated with two polynomials,joint at x/h ≈ 2.5 (see figure 1) and conditioned as to have a continuous first derivative. Then,this polynomial RRI was extrapolated to y/h = −1. Values of LR/h so obtained are listed intable 2. Estimated LR/h of both ramps appear to scale with Reθ , according to a relationship thatStella et al. (2017) modelled with a power-law, in the form:

LR/h ∼ Remθ , (2.1)

where m ≈ −0.1 for Reθ < Reθ,c and m ≈ −0.55 for Reθ > Reθ,c. Even if the cause ofthe change of exponent was not identified in that study, the critical value Reθ,c was evaluatedto approximately 4100. As shown in figure 3, Reθ appears to be more adapted at scaling LR/hacross experiments than Reh, at least for Reθ < Reθ,c. For this reason, in the remainder of thepaper we will systematically consider Reθ as the relevant Reynolds number of the flow.

3. A footprint of the incoming boundary layer

The δe/h effect observed at reattachment (Adams & Johnston (1988a)) suggests that theincoming boundary layer influences the flow after separation. If this is so, it can be expected thatthe separated flow contains a distinctive footprint of the incoming boundary layer, that possiblysurvives up to reattachment. It seems then convenient to start our investigation by identifyingsuch footprint, and by showing that its strenght depends on δe/h. In order to do so, we use the


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〈(u′)2〉+ for the GDR ramp at Reθ = 3617. Symbols:+ hotwire measurements; � PIV data from the R2 ramp auxiliary field; ⋆ PIV data from the GDR rampauxiliary field. DNS at Reθ = 2537 as given in Schlatter & Örlü (2010). For clarity, only one pointevery three is reported for each dataset.

statistical distribution of the Turbulent/Non-Turbulent Interface (TNTI) as a tracer, to highlightchanges of flow properties in the streamwise direction. This choice is based on several TNTIcharacteristics that appear to be well suited to our purposes. Firstly, the TNTI is an inexpensivetracer. Indeed, the TNTI can be detected even on simple 2D2C PIV fields, with no need of extraexperimental or numerical efforts (Chauhan et al. (2014c)). Secondly, in our flow the TNTI existson the entire velocity field, which allows simple assessment of the streamwise evolution of theflow (Stella et al. (2017)). Finally and most importantly, the statistical distribution of the TNTIprovides a reliable boundary layer signature, to which we can compare the separated flow understudy. In particular, we remind that in a turbulent boundary layer the instantaneous TNTI locationabove the wall approximately follows a gaussian distribution, scaled by δ (see references at § 1).

In the following subsections, we investigate the TNTI distribution in the separated flow understudy, to determine wheather the gaussian form typical of the incoming boundary layer survivesto separation. It is expected that TNTI properties will remain similar to those of the incomingboundary layer, as long as the latter has a dominant influence on the separated flow. Comparisonof the two ramps allows to highlight the effects of the parameter δe/h. The TNTI is detected


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Reh/104 3 4 5 6 7 10 13.3 20 26.7

δe/h 0.92 0.86 0.82 0.82 0.91 0.34 0.28 0.33 0.29

uτ/[ms−1] 0.68✝ 0.78 0.93 1.10 1.25 0.7✝ 0.86✝ 1.12 1.57Reτ 1270 1310 1750 2130 2646 840 1090 1456 1622Reθ 2006 3262 4122 4738 5512 1788 2547 3617 4340H12 1.43 1.43 1.40 1.40 1.37 1.57 1.57 1.42 1.40

LR/h 5.42 5.22 5.1 4.79 4.4 5.62 5.49 5.37 5.22

TABLE 2. Characteristic parameters of the flow over the two ramps. The ratio δe/h is computed at x/h ≈ 0,as the distance from the wall where U ≈ U∞. Other boundary layer properties are measured at a referencesection x/h = −9, at which the pressure gradient is zero (see Stella et al. (2017)). Friction velocity uτ isobtained with the composite profile of Chauhan et al. (2009), with the exception of those marked with thesymbol ✝, that were computed with the Clauser chart method (see Wei et al. (2005), among others). It isReh = U∞h/ν, Reτ = δuτ/ν and Reθ = U∞θ/ν. H12 is the shape factor (≡ δ1/θ, where δ1 is thedisplacement thickness). LR/h is computed from a polynomial fit of the mean RRI, detected on the meanstreamwise PIV velocity field (see § 2.3).

following the method proposed by Chauhan et al. (2014c) and adapted to separated flow byStella et al. (2017). Its main steps are reminded hereafter.

3.1. Detection of the TNTI

Following da Silva et al. (2014) and Chauhan et al. (2014b,c), the instantaneous TNTI can beidentified with a threshold on the field of a dimensionless turbulent kinetic energy, defined as:

k =100

9(U2∞

+ V 2∞)

1∑

m,n=−1

[(um,n − U∞)2 + (vm,n − V∞)2]. (3.1)


Reθ Reτ YT /δ σT /δ SkT KtT

Chauhan et al. (2014c) - 14500 0.67 0.11 - -

Present study R2

2006 1300 0.64 0.12 0.36 3.323262 1310 0.60 0.13 0.34 3.124122 1750 0.61 0.13 0.36 3.124738 2130 0.61 0.13 0.38 3.175512 2646 0.62 0.12 0.34 3.16

Present study GDR

1788 874 0.62 0.13 0.32 3.582545 1128 0.62 0.13 0.34 3.203617 1490 0.67 0.11 0.23 3.104340 1883 0.63 0.12 0.25 3.00

TABLE 3. Comparison of statistics of TNTI position in the boundary layer. YT is the mean TNTI positionabove the wall and σT is its standard deviation. SkT = µT3/µT2

3/2 and KtT = µT4/µT22 are the

skewness and the kurtosis coefficients of the TNTI distribution, respectively (see § 3.2). Expected valuesfor a normal distribution are SkT = 0 and KtT = 3.

In eq. 3.1, k is locally averaged on a kernel of side equal to 3 vectors, to smooth out PIV noise.The indexes m and n allow to iterate on the two dimensions of the kernel. The threshold valuekth is computed iteratively from instantaneous PIV snapshots of the incoming boundary layer,captured on the auxiliary PIV fields. Based on several analysis of TNTI statistics in turbulentboundary layers (see Corrsin & Kistler (1955) and Chauhan et al. (2014a) among others), kth ischosen as the smallest k value for which the following condition is verified:

YT + 3σT ≈ δ, (3.2)

where YT is the mean wall-normal TNTI position, σT is its standard deviation and δ is estimatedwith the composite boundary layer profile conceived by Chauhan et al. (2009). kth stronglydepends on free-flow turbulence and PIV noise, but it is relatively insensitive to Reθ . In thecase of the R2 ramp, one can consider kth = 0.365. As for what concerns the GDR ramp, areference threshold value kth ≈ 0.21 can be retained for all Reθ . The mean TNTI is identifiedsimply, by detecting the kth isoline on the mean k field. Values of YT /δ and σT /δ correspondingto kth are listed in table 3. Higher order statistics, defined in the following subsection, are alsolisted for future reference.

3.2. Statistical distribution of the TNTI: definitions

For what follows, it is practical to compute the positions of the TNTI in the local framepresented in figure 4, which generalises the streamwise-wall normal one used for boundary layers(Corrsin & Kistler (1955), Chauhan et al. (2014c)). Let us start by considering the mean TNTI.This is a smooth line, which at separation is placed at YT,e/δe ≈ 0.5, where YT,e is the meanTNTI position above the upper edge of the ramp. After separation, the mean TNTI gently curvestoward the wall. We define a curvilinear abscissa s along the mean TNTI. For simplicity, theorigin of s is placed at x = 0. Let also ns be a normal to the mean TNTI. The instantaneousTNTI location for any given s is defined by dTj(s), the signed distance (positive up) along ns

between the j-th instantaneous TNTI and the mean TNTI. The probability density function ofdTj(s) is then NT (s). To characterise NT (s), we will consider its standard deviation σT , itsskewness coefficient SkT = µT3/µT2

3/2 and its kurtosis coefficient KtT = µT4/µT22. In

these expressions, µTn = E (dT − E (dT ))n is the nth-order central moment of NT , and E is

the expected value. According to Cintosun et al. (2007), σT should be related to the characteristiclarge scale of the flow. Then, it should be σ ∼ δ in those regions in which the footprint of theincoming boundary layer is strong. The coefficients SkT and KtT carry information on the shape


Mean TNTI

Mean RRI

s=0

FIGURE 4. Notations and reference frames for the study of the TNTI distribution in space. dT (s∗) is, theposition of the instantaneous TNTI at the curvilinear abscissa s∗ (defined on the mean TNTI).

of NT (s), respectively on its symmetry and on the size of its tails (i.e. on the presence of outliers,see Westfall (2014)). For a gaussian distribution, it is SkT = 0 and KtT = 3.

3.3. Streamwise evolution of the statistical distribution of the TNTI

The form of NT (s) over the R2 ramp is reported in figure 5, at several streamwise locations.In a large neighbourhood of the mean separation point, the TNTI seems roughly distributed asa gaussian random variable: the statistical properties of the TNTI that are typical of boundarylayers seem then to survive to separation, persisting over a part of the recirculation region. Letus call gaussian length (noted LG) this first part of the separated flow. Further downstream,however, NT (s) deviates progressively from a gaussian distribution. NT (s) is more and moreskewed: its inner (i.e. towards the wall) tail is shortened and the TNTI sample is more con-centrated slightly under the mean TNTI. The streamwise evolution of some TNTI properties isnot surprising: indeed, unlike canonical flows studied in previous works, the massive separationunder investigation is not equilibrated (since the boundary conditions evolve) nor, in general,self-similar (see for example Song & Eaton (2004)). The evolution of NT (s) on the GDR ramp(not reported) is qualitatively comparable to the one observed on the R2 ramp, but LG is foundto be much longer in the latter experiment. The analysis of the streamwise evolution of statisticalmoments allows to sketch a possible explanation for different LG magnitudes. The streamwiseevolutions of NT statistics over both ramps are presented in figure 6. For each ramp, all curvescollapse together nicely. Then, only R2 data at Reθ = 3262 and GDR data at Reθ = 3617 arecompared, because their incoming boundary layers have very similar turbulent states. Figures 6(a), (c) and (e), respectively show the evolutions of σT /δe, SkT and KtT in function of thenon-dimensional streamwise coordinate x∗. It appears that, in a large region downstream ofseparation, all considered TNTI statistics keep values that are very similar to those measuredin the incoming boundary layer, at x/h ≈ −9, with substantial agreement among the tworamps. These quantitative observations confirm that the δ-scaled, gaussian form of NT persistsdownstream of separation, over a domain x ∈ (0, LG). According to figure 6 (a), LG does notscale with LR. Indeed, it is L∗

G ≈ 0.8 on the R2 ramp and L∗

G ≈ 0.2 to 0.3 in the case of the


(a)

dT/σ

T

−3

−2

−1

0

1

2

3(b)

−3

−2

−1

0

1

2

3

(c)

dT/σ

T

−3

−2

−1

0

1

2

3(d)

(e)

dT/σ

T

−3

−2

−1

0

1

2

3

NT

0.1 0.2 0.3 0.4 0.5

(f)

−3

−2

−1

0

1

2

3

NT

0 0.1 0.2 0.3 0.4 0.5

FIGURE 5. Normalised TNTI distribution on the R2 ramp. (a) x∗ = 0.2; (b) x∗ = 0.4; (c) x∗ = 0.6; (d)x∗ = 0.8; (e) x∗ = 1.1; (f) x∗ = 1.4. Symbols: © Reθ = 2006; � Reθ = 3262; � Reθ = 4122; ✚

Reθ = 4738; N Reθ = 5512. gaussian distribution.


��0�S

��0��

T�/

�1 �

b5SK

b5K

b b5K c c5K

T�/

b5SK

b5K

b K cb cK

��0��

��0�ST�/

��

b

c

S

b b5K c c5K

T�/

b

c

S

b K cb cK

��0��

��0�ST�/

��

k

K

3

�δb b5K c c5K

T�/

k

K

3

�1 �

b K cb cK

FIGURE 6. Streamwise evolution of TNTI statistics, represented with different normalisation: (a) and(b) σT /δe; (c) and (d) skewness coefficient SkT = µT3/µT2

3/2; (e) and (f) kurtosis coefficientKtT = µT4/µT2

2. Subfigures in the same column share the same streamwise normalisation: (a),(c) and (e)x/LR; (b), (d) and (f) x/δe. Symbols: � R2 ramp (Reθ = 3262); ⋆GDR ramp (Reθ = 3617). Solid lines( ) show reference values for the gaussian TNTI distribution observed in the incoming boundarylayer, at the reference section x/h ≈ −9 (see 3): (a) and (b) σT /δe ≈ 0.12; (c) and (d) SkT = 0.3; (e)and (f) KtT = 3. A dashed line ( ) is used to highlight common trends.

GDR ramp. Anyway, since it is hGDR/hR2 = 3 and LR/h ≈ 5 for both ramps (at least ifReθ dependencies are neglected), the dimensional value of LG is approximately constant acrossexperiments. Interestingly, on both ramps it is L∗

G ≈ δe/h: since LR ∼ h, it is then tempting toput LG ∼ δe. Figure 6 (b), (d) and (f) appear to support this idea. By normalising the streamwisecoordinate on δe, indeed, trends of all NT parameters collapse at least on the entire extent of L∗

G.As for what concerns the domain x∗ > L∗

G, the footprint of the incoming boundary layerappears to progressively wane. NT deviates from a gaussian distribution, as shown by theincreasing values of both SkT and KtT (figure 6 (c) and (e)). Positive skewness coefficients arecompatible with a longer outer (i.e. toward the free stream) tail and higher values of KtT indicate


a stronger presence of outliers. In addition, σT also increases on both ramps, with approximatelylinear trends. The behaviour of σT reminds the linear increase of the cross-stream scale of theshear layer (see § 4), so that it does not seem unreasonable to associate the domain x∗ > L∗

G

to a certain predominance of the separated shear layer. Anyway, mind that in free shear layersNT has also been observed to be gaussian (Attili et al. (2014)): then, the non-gaussian formfound on x∗ > L∗

G might be indicative of a transition region, possibly toward a new boundarylayer, dominated by the development of the shear layer. All in all, the analysis of the statisticalbehaviour of the TNTI highlights a sizeable footprint of the incoming boundary layer on theseparated flow. Such footprint dominates the flow on a length LG downstream of separation, butprogressively weakens as the separated shear layer develops. More importantly, the parameterδe/h appears to determine the relative strength of the incoming boundary layer with respect tothe development of the separated shear layer. The higher is δe/h, the more the footprint of theincoming boundary layer is persistent.

4. Effects on the development of the separated shear layer

Now that we have identified a clear boundary layer footprint, it is time to get some insightinto its effects on the flow after separation. In this respect, it seems interesting to begin byinvestigating the separated shear layer, because it is one of the main features of the flow.Following Dandois et al. (2007), in a mean bidimensional flow the growth of the separated shearlayer can be simply assessed by considering the streamwise evolution of the vorticity thicknessδω, defined as:

δω(x) =U∞(x) − Umin(x)

(∂U(x, y)/∂y)max

, (4.1)

where U∞ and Umin are the local maximum and the local minimum streamwise velocities. It isUmin(x) < 0 for x < LR and Umin(x) ∼ U∞ in the entire separated flow (see Le et al. (1997)and Dandois et al. (2007)). Figure 7 (a) compares the streamwise evolution of δω/h observedon the two ramps. The streamwise coordinate is once again x∗. It appears that the evolutions ofδω(x)/h collapse on two quasi-parallel, piecewise linear trends (also see Dandois et al. (2007)).The linear growth of the separated shear layer is consistent with the free shear layer approxima-tion, in particular in a large neighbourhood of separation. We indicate with x∗ the streamwiseposition at which the δω/h trend changes its slope. The value of x∗ appears to be at most a weakfunction of δe/h: based on available measurements, we put x∗ = 0.55± 0.05. For convenience,we also consider that x∗ divides the flow domain in a separation region (for x∗ < x∗) and areattachment region (for x∗ > x∗). In what follows, we will focus on the separation region only.

4.1. Two competing length scales

The slopes of the δω/h trends shown in figure 7 (a) correspond to the non-dimensional growthrates of the separated shear layer. On x∗ < x∗, let us indicate this quantity with the symbol γS .It is:

dδωdx

= γS (LR/h)−1

. (4.2)

Figure 7 (a) suggests that γS is relatively insensitive to Reθ effects, and at most a weak functionof δe/h. This is confirmed by direct measurements, reported in figure 7 (b), which give γS =1.1± 0.07 across experiments. If γS can be considered almost constant, then the main differencebetween the R2 ramp and the GDR ramp on x∗ < x∗ is the intercept δω (0) /h. In this regard, it isempirically found that δω (0)R2

≈ δω (0)GDR ≈ 6mm, which immediately suggests that δω(0)does not scale with the height of the ramp h. It suits our purposes to consider that δω (0) = βδe,in which β is a proportionality factor. In these experiments, it is β ≈ 0.21. The value of β


h�2

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)x

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8

84*

847

84d

843

x

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84e

x

x4x

x4*

x45

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x * 5 7 L d

FIGURE 7. Shear layer scaling. (a) Shear layer development, expressed as the streamwise evolution ofvorticity thickness δω . For sake of clarity, only one in each six points is represented in the case of the R2ramp, and only one in fifteen in the case of the GDR ramp. The solid line ( ) indicates the unityslope. For reference, the upper edge of the ramp (x/LR = 0) is visualised by a dotted line ( ) andthe corresponding values of δω/h by dashed lines ( ). Symbols for the GDR ramp (grey online):◭Reθ = 1788; ◮Reθ = 2547; ⋆Reθ =3617;5Reθ = 4340. Symbols for the R2 ramp (white online) asin figure 5. (b) Variation of the product (LR/h) dδω(x)/dx with Reθ . Symbols: ▽ R2 ramp; H GDR ramp;

γS = 1.1. Fine, solid lines indicate the ±0.07 tolerance on γS .

might depend on the maximum velocity gradient near the wall, and hence on the fullness of theboundary layer velocity profile. Based on these considerations, on x∗ < x∗ it does not seemunreasonable to put:

δω(x)

h≈ γSx

∗ + βδeh. (4.3)

Eq. 4.3 has two important implications. Firstly, it appears that δω depends on two characteristiclength scales, which can be intuitively related to different, coexisting phenomena: the h-scaled,linear term relates to the growth of a free-like shear layer after separation (the h scaling, however,is specific to separating/reattaching flows), while the constant δe term shows the persistinginfluence of the incoming boundary layer dowstream of separation. Secondly, the relative weightof these two terms appears to vary in the streamwise direction. In this respect, it is interesting torecast eq. 4.3 as:

δω(x)

h≈ γSx

∗

(

1 +β

γSx∗

δeh

)

. (4.4)

Eq. 4.4 suggests that the main contribution to δω is provided by βδe on a subdomain x∗ <[(β/γS) (δe/h)], the extent of which, indicated with x∗

δ , increases with δe/h. For δe/h = 0, it isalso x∗

δ = 0 and the growth of δω after separation is correctly captured by the free shear layeranalogy. For δe/h ≫ γS/β, instead, the influence of the boundary layer should cover the entirerecirculation region. In this case, it is thought that the flow might be better approximated by aboundary layer on a rough wall, rather than by a free shear layer. Then, for asymptotic values ofδe/h the flow has only one characteristic length scale, either h or δe. For intermediate values ofδe/h, anyway, the two length scales are in competition: then, the flow in the separation regionmight evolve from a pure h scaling, to a mixed scaling based on h and δe, to a scaling dominatedby δe, as δe/h increases. It is pointed out that good collapse of δω/h on the entire recirculationregion (figure 7) suggests that an expression containing a similar dependency on both h and δe


might also exist for x∗ > x∗, so that present considerations might be extended, at least to adegree, to the reattachment region. It is stressed that these results are in good agreement withfindings reported at § 3: δe/h provides a measure of the competition between the influence ofthe incoming boundary layer and the development of a free-like shear layer originating from theupper edge of the ramp. According to this idea, the higher is δe/h, the further downstream theinfluence of the boundary layer persists after separation. In this instance, we report that availabledata give x∗

δ < 0.06 on the GDR ramp, and x∗

δ < 0.16 on the R2 ramp. Interestingly, it isL∗

G/x∗

δ ≈ 5 for both experiments, and hence L∗

G ∼ x∗

δ . Then, it seems possible to correlate theregion in which the term βδe dominates eq. 4.4 to a strong boundary layer footprint.

4.2. Effects on mean shear

The multi-scale dependency of δω expressed in eq. 4.4 has direct consequences on the velocitygradient ∂U/∂y, which in the bidimensional flow under investigation provides the main compo-nent of mean shear. By recasting eq. 4.1 and making use of eq. 4.4, one obtains the followingexpression:

∂U(x, y)

∂y

∣

∣

∣

∣

max

=U∞(x) − Umin(x)

δω≈

U∞

γShx∗

(

1 +β

γSx∗

δeh

)

−1

. (4.5)

According to eq. 4.5, the presence of an incoming boundary layer increases the characteristiclength scale of ∂U/∂y, without fundamentally changing its velocity scale. Then, the higheris δe/h, the less intensely the flow in the separation region is sheared. Due to the centralrole of ∂U/∂y, this suggests that a variation of δe/h will have far reaching consequenceson the separated flow. In particular, lower mean shear is likely to induce lower turbulentproduction, and hence less intense Reynolds stresses in the whole separated flow. Interestingly,Adams & Johnston (1988a) and Stella et al. (2017) show that turbulent shear provides thestrongest contribution to the pressure gradient at reattachment, so that the reduction of ∂U/∂ydue to δe/h might indeed be at the origin of the progressive decrease of reattachment pressureobserved by those authors (see also § 1). This topic, clearly connected to the matter of this paper,is currently being investigated.

5. Shear layer growth and mass entrainment

The previous section highlighted some important effects of δe/h > 0 on the flow in theseparation region. Anyway, figure 7 and eq. 4.4 show that the growth of the separated shearlayer is approximately linear, regardless to its strenght relative to the boundary layer footprint.If this is so, the free-shear layer analogy proposed by several researchers appears to still hold. Itseems then possible to rely on one of the cornerstones of this analogy, that is that the growth ratedδω/dx depends on how mass in entrained into the shear layer. According to Pope (2000), onecan put:

dδωdx

∼∑

v∗E , (5.1)

were∑

v∗E is the total mean mass entrainment rate through the boundaries of the separated shearlayer. For each of those boundaries, v∗E = vE/U∞ is a mean, large-scale mass entrainment rate.

Implications of eq. 5.1 need to be assessed in the light of findings at § 4. Indeed, whiledδω/dx appears to be almost insensitive to δe/h, mass entrainment, driven by the velocity field,is certainly affected by changes in mean shear. Then, it does not seem unreasonable to expectthat the proportionality factor between the two terms of eq. 5.1 will be a function of δe/h.

In this section we aim at completing our investigation of the role of δe/h by shedding somelight into this matter. To this end, it seems convenient to start by computing the mean mass


FIGURE 8. Control volume including the separated shear layer at Reθ = 3617.

balance over the shear layer, to provide a global characterisation of mass exchanges. Then, wewill analyse local mass fluxes, which lead to the computation of mass entrainment rates.

5.1. Mean mass balance

A first necessary step toward the computation of the mean mass balance consists in defininga control volume representative of the shear layer. According to Stella et al. (2017), one validchoice is a volume Vc delimited by two vertical segments, placed at the positions of the meanseparation point (called inlet) and mean reattachment point (called outlet); by the mean TNTI,which separates the shear layer from the free flow; and by the mean RRI. For the sake of example,the control volume Vc for the GDR ramp flow at Reθ = 3617 is shown in figure 8. Consideringthat the mean field is bidimensional, the total mass flux per spanwise unit length through each ofthe sides of Vc is given by:

mj = −ρ

∫

Sj

Ui(s)ni(s)ds = −ρ

∫

Sj

(U(s) sin(φ(s)) + V (s) cos(φ(s))) ds, (5.2)

where Sj is the length of one side, s a curvilinear abscissa, n(s) is the local normal to Sj

(pointing outward of Vc) and φ is the angle between n(s) and the Y axis. The index j goes from1 to 4. j = 1 indicates the inlet at the mean separation point; then, j increases counterclockwise,so that j = 4 identifies the mean TNTI. Of course, continuity implies that

∑

mj = 0.Measured mean mass fluxes are reported in table 4. Uncertainties on mass balance are mainly

caused by corrupted velocity vectors produced by laser reflections on the wall, in particular ina neighbourhood of the mean separation point. As already pointed out in Stella et al. (2017),m2 must be zero, because in average the backflow balances shear layer entrainment throughthe RRI in a neighbourhood of the mean separation point (see Chapman et al. (1958) andAdams & Johnston (1988b)). Our results also agree with Stella et al. (2017) in evidencing thatm4 is not negligible. Indeed, since the TNTI is not a streamline, mass entrainment through theTNTI compensates the difference of mass fluxes between the outlet and the inlet, i.e. m4 =−m3 − m1 6= 0. The role of the TNTI appears to be even stronger on the GDR ramp than onthe R2 ramp: while on the latter the TNTI contributes to mass balance with approximately 30%of the mass injected into Vc by the incoming boundary layer, on the former m4 becomes thedominant positive mass contribution.

The decreasing weight of the incoming boundary layer on the global mass balance seemsconsistent with a lower value of the parameter δe/h, as follows. It appears from table 4 that themass flow at the inlet m1 does not scale with ρhU∞. Of course, a straightforward alternativeis scaling m1 on ρδeU∞, as reported in table 5. This second normalisation appears to be morerelevant: it is m1/ρδeU∞ ≈ 0.5± 0.05. The value of this ratio is of the same order of magnitudeof YT,e/δe. Although scatter is not always negligible, it seems then possible to assume that m1


R2 GDR

Reθ 2006 3262 4122 4738 5512 1788 2547 3617 4340

m∗

1 0.39 0.43 0.41 0.45 0.44 0.17 0.15 0.15 0.14m∗

3 -0.53 -0.56 -0.55 -0.64 -0.59 -0.45 -0.47 -0.45 -0.49m∗

4 0.14 0.14 0.14 0.20 0.17 0.27 0.28 0.27 0.33

ǫ∗m = (m∗

1 + m∗

3 + m∗

4) 0.00 0.00 0.007 0.006 0.019 -0.03 -0.04 -0.03 -0.02

m∗

2 -0.005 -0.007 -0.007 -0.004 0.005 -0.01 -0.005 0.008 0.014

TABLE 4. Mass fluxes normalized on ρhU∞. The error ǫ∗m does not include m∗

2 because m∗

4 shouldbalance the difference between the inlet and the outlet, independently of how well the m∗

2 = 0 condition ismet.

R2 GDR

Reθ 2006 3262 4122 4738 5512 1788 2547 3617 4340

m1/ρδeU∞ 0.45 0.50 0.50 0.55 0.48 0.48 0.56 0.45 0.54

TABLE 5. Inlet mass fluxes (m1) normalized on ρδeU∞.

is sized by YT,e, and more in general by the outer scales of the incoming boundary layer. Then,the amount of mass transported by the incoming boundary layer into Vc is comparable in thetwo experiments, because δe is also approximately the same. However, table 4 also suggeststhat more mass leaves from the outlet as h increases, that is, at least in our experiments, fordecreasing values of δe/h. Since the mean separated shear layer is stationary, this implies that,to verify continuity, the mass contribution of the TNTI must also increase as δe/h decreases.

5.2. Local mean mass fluxes

Now that some of the effects of δe/h on the mean mass balance have been identified, it isconvenient to consider the local mass fluxes along the RRI and the TNTI. The normalised localflux at any point of the two interfaces can be estimated as:

m∗

xi =1

ρU∞

dmi

ds= −

1

U∞

(U(s) sin(φ(s)) + V (s) cos(φ(s)). (5.3)

This finer analysis can provide information which is hidden by the integral approach of § 5.1,such as the spatial distribution of mass fluxes and their scaling laws.

5.2.1. RRI fluxes

The streamwise evolution of m∗

x2 is reported in figure 9(a) for Reθ = 3262 (R2 ramp) andReθ = 3617 (GDR ramp). The distribution of m∗

x2 along the mean RRI is approximately odd,with a change of sign at x∗ ≈ x∗. As such, m∗

x2 appears to be very well correlated to thedevelopment of the separated shear layer (see figure 7 (a)). Curves from both experimentscollapse together nicely, with the exception of the domain x∗ > 0.7. Since mass entrainmentat reattachment appears to be correlated to turbulent shear (Stella et al. (2017)), this differencemight be a consequence of the different value of δe/h (also see pressure distributions andturbulent shear stress profiles in Adams & Johnston (1988a)).

Figure 9 (a) proves that, even if m∗

2= 0, local mass transfer through the RRI is not negligible.

On x∗ < x∗, m∗

x2 injects mass into Vc: we will indicate quantities relative to this domain with


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FIGURE 9. Normalized local mass fluxes along the mean interfaces of the separated shear layer at for Reθ =3262 (R2 ramp - � ) and Reθ = 3617 (GDR ramp - ⋆ ). (a) m∗

x2 along the RRI. Only one in each six pointsis represented in the case of the R2 ramp, and only one in fifteen in the case of the GDR ramp. (b) m∗

x4

along the TNTI. Only one in each ten points is represented in the case of the R2 ramp, and only one intwenty in the case of the GDR ramp. Red arrows indicate the effect of increasing δe/h.

the symbol IN . On the contrary, m∗

x2 extracts mass from Vc on x∗ > x∗: quantities relativeto this domain will be marked by the symbol OUT . Considering the scaling of figure 9(a), theentrainment rate on, say, x∗ < x∗ is simply given by:

v∗E,RIN = −

1

ρU∞S

∫ s

0

ρUi(s)ni(s)ds, (5.4)

where s is the curvilinear abscissa at x∗ = x∗ and:

S =

∫ s

0

ds. (5.5)

v∗E,ROUT can be computed with an expression similar to eq. 5.4. Available data give v∗E,R

IN ≈

0.0223±0.002 and v∗E,ROUT ≈ −0.0225±0.0015on the GDR ramp; v∗E,R

IN ≈ 0.0224±0.002

and v∗E,ROUT ≈ −0.0205± 0.0015 on the R2 ramp. These measurements clearly indicate that

the mean entrainment rate through the RRI is independent of δe/h, in spite of the influence ofthis latter parameter at reattachment. In general, v∗E,R is also weakly affected by other parameterssuch as h, Reh (at least if Reh > 36000, see Nadge & Govardhan (2014)), the value of Reθ in theincoming boundary layer (see table 6) and, to a large extent, LR (see Stella et al. (in press)).

5.2.2. TNTI fluxes

Figure 9(b) presents the streamwise evolution of m∗

x4, obtained by applying eq. 5.3 to theTNTI. As in the case of the RRI, the behaviour of m∗

x4 changes at x∗ ≈ x∗, with sizeablymore intense transfer in the reattachment region. In this respect, comparison with Stella et al.

(2017) suggests that the peak of mass entrainment through the TNTI is reached in proximity ofthe position of maximum pressure gradient. Once again, the two ramps show different trendsin this region, which might be related to δe/h. It is evident that the intensity of local fluxesis higher in the case of the GDR ramp. This seems consistent with the increased contributionbrought by the TNTI to mass balance, as δe/h decreases (see § 5.1). Anyway, the reattachmentregion accounts for 80% to 90% of m4 on the R2 ramp, but for only 60% to 80% of m4 on


R2 GDR

Reθ 2006 3262 4122 4738 5512 1788 2547 3617 4340

v∗E,T /10−2 0.62 0.86 1.08 1.57 1.85 1.79 1.94 1.88 2.62

v∗E,RIN/10−2 2.26 2.24 2.17 2.43 2.11 2.07 2.22 2.40 2.23

TABLE 6. Mean entrainment rates at the mean TNTI and at the mean RRI on x∗ ∈ (0, x∗).

the GDR ramp. Then, mass entrainment through the TNTI is fundamentally concentrated in thereattachment region in the case of the R2 ramp, while it acts more homogenously over the GDRramp. The mean entrainment rate on x∗ ∈ (0, x∗), indicated with v∗E,T , can be computed simply,

by adapting eq. 5.4 to the TNTI. Table 6 shows that, unlike v∗E,RIN , v∗E,T increases with Reθ. In

addition, the two ramps differ by the relative weight of v∗E,T and v∗E,RIN . Generally speaking,

v∗E,T is of the same order of magnitude as v∗E,RIN on the GDR ramp, but it is sizeably lower

on the R2 ramp. This might appear counterintuitive, at first sight, as one could expect v∗E,T torise accordingly to the strenght of boundary layer footprint. Anyway, it should be reminded that∂U/∂y decreases as δe/h increases. This tends to hinder turbulent production, which reducesmixing and hence the intensity of transfer among different regions of the flow.

5.3. Total entrainment rates and shear layer growth

In the previous paragraphs we showed that the parameter δe/h sizeably affects mass entrain-ment from the free-stream, while leaving entrainment from the reverse flow unchanged. Let usnow get back to the correlation between dδω/dx and

∑

v∗E = v∗E,RIN + v∗E,T . As expected,

figure 10 shows that eq. 5.1 is well verified in both experiments. Findings reported at § 5.2.1suggest that the variation of

∑

v∗E is fundamentally driven by v∗E,T . Considering that γS isapproximately constant and that LR/h is a function of Reθ (see figure 3 (b)), eq. 4.2 clearlyindicates that Reθ determines, for each ramp, the position of available datapoints along the lineartrends of figure 10. This confirms that, at least if other parameters such as the turbulent state of theincoming boundary layer or the geometry of the ramp are kept constant, the proportionality factorbetween the two terms of eq. 5.1 (i.e. the slopes of the linear trends of figure 10) is mainly affectedby δe/h. Our next objective is to investigate whether such trends can be collapsed together by ascaling factor that takes into account the effects of δe/h. Since it was observed that δe/h changesmass entrainment from the free stream, a mean mass balance seems a promising starting pointfor our discussion. Unlike at § 5.1, we now focus on the separation region exclusively, i.e. onx∗ < x∗. With reference to figure 11, let us define a new control volume VC2, that correspondsto the portion of VC for x∗ < x∗. To begin with, we can directly write:

mT + mR = mx − m1, (5.6)

where mT , mR and mx are the norms of mass fluxes through the TNTI, the RRI and the outletof VC2, respectively. According to table 5, it is straightforward to put:

m1 ≈ ρU∞

δe2. (5.7)

In principle, computing mx requires to know the shear layer velocity profile at x∗ = x∗,indicated with U (x∗, s). A tempting starting point to estimate U (x∗, s) is the velocity profileof the free shear layer, which is usually approximated by an error function. For small values ofδe/h and low turbulent intensities, Tanner (1973) (among others) shows that this profile can beadapted to massively separated flows and provide good predictions of mean flow properties, such


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/δ/8

/δ/�

/δ/

/δ/�

/δ/�

��ω��

/δx� /δ8 /δ88 /δ8

FIGURE 10. Correlation between dδω/dx and∑

v∗E (eq. 5.1). Symbols: ▽ R2 ramp; H GDR ramp;linear best fits in the form y = 1/Kω x, with 1/Kω being the slope. The hatched area in (a)

indicates v∗E,RIN .

ph

FIGURE 11. View of the control volume VC2, that covers the separation region. m1, mR, mT and mx

are the mass fluxes through the inlet, the RRI, the TNTI and the outlet, respectively. The RRI is adoptedas lower boundary of shear layer development. The upper boundary is indicated with a dashed line. Redlines are used to represent streamwise velocity profiles at the inlet and at the outlet. The insert details thedefinitions of Uc and ∆m.

as reattachment wall-pressure. However, these hypotheses are not generally acceptable in thepresent framework, as they defeat the very purpose of our discussion. An alternative approachto estimate mx might be based on general considerations on the topology of the flow. To beginwith, if U (x∗, s) = U∞, the amount of mass entrained by the flow through the outlet section of


VC2, indicated with Sx∗ , would be simply given by:

mU∞= ρU∞Sx∗ . (5.8)

In itself, mU∞is not a good estimate of mx, because ∂U/∂y is not negligible along a wide

portion of Sx∗ . Then, it does not seem unreasonable to introduce a mass entrainment deficit,representing the amount of mass that does not cross Sx∗ due to the velocity gradient. Eq. 5.7suggests that the velocity scale is approximately U∞ in the outer part of Sx∗ . Then, the verticalvelocity gradient mainly depends on the separated shear layer and, based on findings at § 4, wecan tentatively put:

∆m ≈ ρ (U∞ − Uc) δω (x∗) , (5.9)

in which Uc is a characteristic convection velocity. Under the free-shear layer analogy, we canrefer to Pope (2000) and define Uc as:

Uc =U∞ + Umin

2. (5.10)

This expression needs to be corrected, to take into account that the lower boundary of VC2 isplaced at U = 0, rather than U = Umin. It is then Uc ≈ U∞/2 and hence:

∆m ≈ ρU∞

2δω (x∗) . (5.11)

With these results, we can propose the following formulation for mx:

mx ≈ mU∞−∆m ≈ ρU∞

[

Sx∗ −δω (x∗)

2

]

. (5.12)

All terms in this expression are known, with the exception of the length of the outlet section.According to figure 11, Sx∗ can be estimated with three terms, as follows:

Sx∗ ≈δe2

+ ph−∆δe. (5.13)

The half-thickness of the incoming boundary layer δe/2 is linked to YT,e. The term ph, with0 < p < 1, takes into account the development of the separated shear layer toward the wall.Measurements on available velocity fields give p ≈ 0.45± 0.05. Finally, the term ∆δe takes intoaccount the inclination of the mean TNTI toward the wall. This term cannot be predicted simply,but for a first approximation we can use geometrical consideration on the shape of VC2 to put:

∆δe/δe ≈ ΦTNTI x∗ LR/h (δe/h)

−1, (5.14)

in which ΦTNTI is the slope of the mean TNTI, approximated with a straight line over the extentof VC2. The investigation of the behaviour of ΦTNTI is beyond the scope of this work, butpreliminary measurements seem to show that:

ΦTNTI (LR/h) ≈ kΦ, (5.15)

where it is kΦ ≈ 0.10 on the R2 ramp and kΦ ≈ 0.03 on the GDR ramp. Surprisingly, the productkΦ (δe/h)

−1 is almost constant across experiments, so that ∆δe/δe varies within a small rangeapproximated by (0.0675± 0.0025). By injecting these results into eq. 5.12 and by making useof eq. 4.3, one finds:

mx ≈ ρU∞

[

δe

(

1− β

2−

∆δeδe

)

+ h

(

p−γS x∗

2

)]

. (5.16)


Let us now get back to eq. 5.6. By plugging in eq. 5.7 and eq. 5.16, simple manipulations leadto:

mT + mR ≈ρU∞

2hγS

(

2p

γS− x∗

)[

1−β + 2∆δe/δe2p− γS x∗

δeh

]

, (5.17)

in which terms were reorganised as to make the dependency on δe/h explicit. We indicate withthe symbols ST and SR the portions of the TNTI and of the RRI, respectively, that delimit VC2.If it assumed that ST ≈ SR ≈ LR/2, it is:

2mT + mR

ρU∞LR≈

∑

v∗E . (5.18)

By making use of this result, eq. 5.17 becomes:

∑

v∗E ≈h

LRγS

(

2p

γS− x∗

)[


δeh

]

. (5.19)

According to eq. 4.2, it is dδω/dx = γS (LR/h)−1. Then, it is possible to write:

∑

v∗E

[


δeh

]

−1

≈dδωdx

(

2p

γS− x∗

)

. (5.20)

If δe/h is sufficiently small, a Taylor expansion can be used to rewrite eq. 5.20 as:

∑

v∗ECh,δqS ≈

1

Kω

dδωdx

. (5.21)

In this expression, it is Ch,δ = (1 + δe/h), qS = (β + 2∆δe/δe) / (2p− γS x∗) and 1/Kω =

(2p/γS − x∗). Available data give qS ≈ 1.17 ≈ 7/6 across experiments. The proportionalityfactor 1/Kω is independent of δe/h, which suggests that 1/Kω might be assimilable to theproportionality factor typical of free shear layers. For a free shear layer with similar values ofU∞ and Umin Pope (2000) predicts:

1

Kω,0≈ 0.24

U∞ − Umin

U∞

≈ 0.28, (5.22)

where, following Le et al. (1997) and Dandois et al. (2007), it was considered Umin/U∞ ≈

−0.15. Significantly, available data give 1/Kω ≈ 0.27. This result agrees very well with theexpected value for free shear layer 1/Kω,0. In this respect, it is interesting to obtain a secondestimation of 1/Kω, by recasting eq. 5.21 as:

1

Kω≈

∑

v∗ECh,δqS

(

dδωdx

)

−1

. (5.23)

Figure 12 shows the evolution of eq. 5.23 with Reθ. Most datapoints appear to be clusteredwithin 1/Kω ≈ 0.31± 0.03, which is once again in good accordance with 1/Kω,0 and with theprediction 1/Kω ≈ 0.27 provided by eq. 5.21. These considerations suggest that eq. 5.21 is quiteeffective at scaling shear layer behaviours across experiments. Then, it seems possible to put:

∑

v∗ECh,δ7/6 ≈

1

Kω,0

dδωdx

. (5.24)

This expression predicts that, in the separation region, the effect of δe/h on rates of massentrainment into the separated shear layer are scaled by a power law of the scaling factor Ch,δ .Once such effects are compensated for, the relationship between mass entrainment and shearlayer collapse on a trend typical of free shear layers.


�v��(��ωv4-3v/��-��6x0

1).9

1).4

1).e

1)20

1)22

1)29

1)24

1)2e

1)50

��-012

0 . 2 5 9 3

FIGURE 12. Evolution of the ratio∑

v∗ECh,δ7/6 (dδω/dx)

−1 (eq. 5.23) with Reθ . Symbols: ▽ R2 ramp;H GDR ramp; best fits in the form y = 1/Kω . Fine, solid lines indicate the ±0.03 tolerance on1/Kω .

6. Conclusions

In this work we experimentally investigated how the incoming boundary layer affects amassively separated turbulent flow. Since some of these effects have already been identifiedat reattachment, we focused on a large neighbourhood of separation. The chosen study casewas the massively separated turbulent flow generated by a sharp edge, 25◦ descending ramp.Significantly, this work could rely on two experimental models with sizeably different values oframp height h (their ratio was 1:3), but otherwise substantially similar geometries and incomingflows. This allowed us to compare the effects of two very different δe/h ratios, in which δe is thefull thickness of the incoming boundary layer at separation.

By using the Turbulent/Non-Turbulent Interface (TNTI) as a tracer, we showed that theincoming boundary layer leaves a clear footprint on the separated flow. In particular, the statisticaldistribution of the TNTI keeps the gaussian form typical of boundary layers on an extent LG

after separation. In both experiments, it is LG/LR ≈ δe/h. Since LR is the characteristic scaleof shear layer development, the ratio δe/h seems to be representative of the relative strength ofthe incoming boundary layer with respect to the separated shear layer.

On these bases, we set out to understand how the footprint of the boundary layer affects theseparated flow. In particular, we considered shear layer development, classically assessed with avorticity thickness δω. We found that shear layer growth remains linear and that dδω/dx appearsto be at most a weak function of δe/h, at least on the δe/h range covered in this study. Anyway,depending on the value of δe/h, the separated flow might pass from a pure h scaling, to a mixedh-δe scaling, to a δe-dominated scaling. In this latter case, we argue that the flow might be betterinterpreted as a thick boundary layer on a rough wall. In addition, we used simple dimensionalconsiderations to show that the higher is δe/h, the less the mean separated flow is sheared.

It is generally admitted that dδω/dx is proportional to the total mass entrainment rate towardthe shear layer. For this reason, in the last part of this work we analysed how δe/h affects massentrainment in the separation region. With a simple mass balance, we showed that the higher isδe/h, the lower is the entrainment rate from the free flow, which seems consistent with a decrease


of mean velocity gradients across different regions of the flow. Anyway, we demonstrated that apower law of the scaling factor Ch,δ = (1 + δe/h) can be used to scale the effects of δe/h onmass entrainment rates. By doing so, the relationship between mass entrainment rates and shearlayer growth appears to collapse on a trend typical of free shear layers.

All in all, this work suggests that separating/reattaching flows assimilable to the one understudy might be thought of as the result of the competition of at least two simpler flows: afree-like separated shear layer, scaled by h, and the incoming boundary layer, scaled by δe,their equilibrium being determined by δe/h. At least to a certain extent, it appears possible toreduce these non-trivial, multi-scale flows to one or the other of their canonical components, byintroducing simple scaling factors based on δe/h, such as Ch,δ . These findings might indicatethat the optimal solutions for controlling or predicting such separating/reattaching flows mightstrongly depend on the parameter δe/h. In future works, we will try to confirm this viewwith wider parametric studies, possibly based on numerical simulations. In particular, it seemsimportant to span a wider Reθ range, as well as to test our results against variations of geometricalparameters such as ramp profile and ER.

Acknowledgments

This work was supported by the French National Research Agency (ANR) through the In-

vestissements d’Avenir program under the Labex CAPRYSSES Project (ANR-11-LABX-0006-01), and by the CNRS through the collaborative project Groupement de Recherche 2502.

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